ff
`
`PaUI HorOWitZ HARVARD UNIVERSITY
`
`Winfield Hi“ ROWLAND INSTITUTE FOR SCIENCE. CAMBRlDGE. MASSACHUSETTS
`
`UNIVERSITY PRESS
`
`CAMBRIDGE
`
`Invensys Ex. 2019
`'
`.
`MICI'O Motlon v. Invensys IPR 2014—00393, page 1
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`

`

`
`
`2. Electronic circuit design.
`1. Electronics.
`Winfield.
`II. Title.
`TK78155.H67
`1989
`621.381 — dc19
`
`1. Hill,
`
`89—468
`CIP
`
`
` Published by the Press Syndicate of the University of Cambridge
`The Pitt Building, Trumpington Street, Cambridge CB2 lRP
`40 West 20th Street, New York, NY 10011—4211, USA
`
`
`10 Stamford Road, Oakleigh. Victoria 3166, Australia
`
`© Cambridge University Press 1980, 1989
` First published 1980
`
`Second edition 1989
`Reprinted 1990 (twice), I991, 1993
`
`
`Printed in the United States of America
`
` Library of Congress Cataloging—in-Publication Data
`Horowitz, Paul, 1942—
`
`
`The art of electronics / Paul Horowitz, Winfield Hill. — 2nd ed.
`p.
`cm.
`Bibliography: p.
`
`Includes index.
`ISBN 0—521-37095—7
`
`
`
`
`
`
`
`British Library Cataloguing in Publication Data
`Horowitz, Paul. 1942—
`The art of electronics. — 2nd ed.
`1. Electronic equipment
`I. Title.
`II. Hill, Winfield
`621.381
`
`ISBN 0'521-37095-7 hardback
`
`
`
`Invensys Ex. 2019
`Micro Motion v. Invensys [PR 2014-00393, page 2
`
`

`

`SIGNALS
`1 .07 Sinusoidal signals
`
`15
`
`
`
`//
`devices to use;
`this fact, combined with
`stunning improvements in transistors, has
`made tunnel diodes almost obsolete.
`The subject of negative resistance will
`come up again later,
`in connection with
`active filters. There you will see a circuit
`called a negative-impedance converter that
`can produce (among other things) a pure
`negative resistance (not just incremental).
`It is made with an operational amplifier
`and has very useful properties.
`
`
`
`
`SIGNALS
`
`A later section in this chapter will deal with
`capacitors, devices whose properties de-
`pend on the way the voltages and currents
`in a circuit are changing. Our analysis of
`dc circuits so far (Ohm’s law, Thévenin
`equivalent circuits, etc.) still holds, even
`if the voltages and currents are changing
`in time. But for a proper understanding of
`altemating-current (ac) circuits, it is useful
`to have in mind certain common types of
`signals, voltagesthat change in time in a
`pafticular way.
`
`
`Figure 1.16
`
`a tunnel diode can actually be an ampli er
`(Fig. 1.16). For a Wiggly voltage vsig, the
`voltage divider equation gives us
`
`
`where n is the incremental resistance of
`the tunnel diode at the operating current,
`and the lower-case symbol vsis stands for a
`small-signal variation, which we have been
`calling AVgig up to now (we will adopt
`this widely used convention from now on).
`The tunnel diode has ”(incr) < 0. That is,
`
`0
`<
`(orv/i)
`AV/AI
`from A to B on the characteristic curve.
`If Tt(incr) z R, the denominator is nearly
`zero, and the circuit amplifies.
`Vbatt
`provides the steady current, or bias,
`to
`bring the operating point into the region of
`negative resistance. (Of course, it is always
`necessary to have a source of power in any
`device that amplifies.)
`A postmortem on these fascinating de-
`vices: When tunnel diodes first appeared,
`late in the 19505, they were hailed as the
`solution to a great variety of circuit prob-
`lems. Because they were fast, they were
`supposed to revolutionize computers, for
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`it
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`le
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`me-
`we
`3m
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`sis-
`106:
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`instance. Unfortunately, they are difficult
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`1 .07 Sinusoidal signals
`Sinusoidal signals are the most popular
`signals around;
`they’re what you get
`
`out of the wall plug.
`If someone says
`something like “take a 10 microvolt signal
`
`at
`l megahertz,” he means a sine wave.
`
`Mathematically, what you have is a voltage
`described by
`
`
`
`V = A sin 2% f t
`
`where A is called the amplitude, and f
`
`isrithe freguency in cycles per secondE
`
`hertz. A sine wave looks like the wave
`
`shown in Figure 1.17. Sometimes it
`is
`
`important to know the value of the signal
`
`at some arbitrary time t = O, in which case
`you may see a phase 4) in the expression:
`V = Asin(27rft + (1))
`
`
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`Invensys Ex. 2019
`.
`.
`Micro Motion v. Invensys IPR 2014-00393, page 3
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`

`

` FOUNDATIONS
`
`Chapter 1
`
`
`
`
`
`if needed.
`with carefully built circuits,
`Higher frequencies, e.g., up to 2000MHz,
`
`
`can be generated, but they require special
`
`
`transmission-line techniques. Above that,
`
`
`you‘re dealing with microwaves, where
`conventional wired circuits with lumped
`circuit elements become impractical, and
`exotic waveguides or “striplines” are used
`instead.
`
`
`
`Figure 1.17. Sine wave of amplitude A and
`frequency f
`
`
`
`
`
`
`
`
`
`The other variation on this simple
`theme is the use of angular frequency,
`which looks like this.
`
`V = A sin wt
`
`
`
`Here, w is the angular frequency in
`radianspersecond Just remember the
`important relation w-—- 27rf and you won’t
`go wrong.
`
`mm
`the cause of their perennial popularity)
`is the fact that they are the solutions to
`certain linear differential equations that
`happen to describe many phenomena in
`nature as well as the properties of linear
`circuits. A linear circuit has the property
`that its output, when driven by the sum
`of two input signals, equals the sum of its
`individual outputs when driven by each '2:
`input signal in turn; i.e., if 0(A) represents
`the output when driven by signal A, then
`a circuit is linear if 0(A + B) = 0(A) +
`0(B). A linear circuit driven by a sine
`wave always responds with a sine wave,
`although
`in
`general
`the
`phase
`and
`amplitude are changed. No other signal
`can make this statement.
`It is standard
`practice, in fact, to describe the behavior
`of a circuit by itsflfryequency response, the
`way it alters the amplitude of an applied
`sine wave as a function of frequency. A
`high-fidelity amplifier, for instance, should
`be characterized by a “flat” frequency
`response over the range 20Hz to 20kHz,
`at least.
`The sine-wave frequencies you will
`usually deal with range from a few hertz to
`a few megahertz. Lower frequencies, down
`to 0.0001Hz or lower, can be generated
`
`1.08 Signal amplitudes and decibels
`
`In addition to its amplitude, there are sev-
`eral other ways to characterize the magni-
`tude of a sine wave or any other signal.
`You sometimes see it specified by peak-t0-
`peak amplitude (pp amplitude), which is
`just what you would guess, namely, twice
`the amplitude The other method is to
`give the root-mean--square amplitude (rms
`amplitude), which is Vrms—— (1/fi)A=
`0.707A (this1s for sine waves only; the ra-
`tio of pp to rms will be different for other
`waveforms). Odd as it may seem, this is
`the usual method, because rms voltage is
`what’ 5 used to compute power The volt-
`age across the terminals of a wall socket (in
`the United States) is 117 volts rms, 60Hz.
`Theamplitude1s 165 velts~(33Q volts pp).
`
`Decibels
`
`How do you compare the relative ampli-
`tudes of two signals? You could say, for
`instance, that signal X is twice as large as
`signal Y. That’s fine, and useful for many
`purposes. But because we often deal with
`ratios as large as a million, it is easier to
`use a logarithmic measure, and for this we
`present the decibel (it’s one-tenth as large
`as something called a bel, which no one
`ever uses). By definition, the ratio of two
`signals, in decibels, is
`
`A
`dB = zoiog10 231
`
`where A1 and A2 are the two signal ampli-
`tudes. So, for instance, one signal of twice
`the amplitude of another is +6dB relative
`
`Invensys Ex. 2019
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`MICI‘O Motion v. Invensys IPR 2014-00393, page 4
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`
`i.e.,
`INI = <NN*>%
`simply obtained by multiplyinE‘by‘le
`complex conjugate and taking the square
`root.
`The magnitude of the product
`(or quotient) of two complex numbers is
`simply the product (or quotient) of their
`magnitudes.
`The real (or imaginary part 0 a com-
`plex number is sometimes written
`real part of N = ’Re(N)
`imaginary part of N = Im(N)
`
`
`MATH REVIEW
`
`APPENDIX B
`
`Some knowledge of algebra and trigonom-
`etry is essential to understand this book.
`In addition, a limited ability to deal with
`complex numbers and derivatives (a part
`of calculus) is helpful, although not en-
`tirely essential. This appendix is meant as
`the briefest of summaries of complex num-
`bers and differentiation. It is not meant as
`a textbook substitute. For a highly read-
`able self—help book on calculus, we rec-
`ommend Quick Calculus, by D. Kleppner
`and N. Ramsey (John Wiley & Sons,
`1972).
`
`I
`
`w COMPLEX NUMBERSW‘]
`
`
`A complex number is an object of the form
`
`N=a+bi
`
`where a and b are real numbers and i
`(called j in the rest of the book, to avoid
`confusion with small-signal currents) is the
`square root of -1; a is called the real
`part, and b is called the imaginary part.
`Boldface letters or squiggly underlines are
`sometimes used to denote complex num-
`bers. At other times you’re just supposed
`to know!
`Complex numbers can be added, sub-
`tracted, multiplied, etc., just as real num-
`bers:
`
`a+bi _ (a+bi)(c—d2’)
`c + di
`(c + di)(c —— di)
`bc — ad _
`ac + bd
`= 2
`2 + 2
`21
`c + d
`c + d
`
`in the
`these operations are natural,
`All
`sense that you just treat 2' as something
`that multiplies the imaginary part, and
`go ahead with ordinary arithmetic. Note
`that i2 = —1 (used in the multiplication
`example) and that division is simplified
`by multiplying top and bottom by the
`complex conjugate,
`the number you get
`by changing the sign of the imaginary
`part. The complex conjugate is sometimes
`indicated with an asterisk. If
`
`N=a+bi
`
`then
`
`N"=a—-bi
`
`The magnitude (or modulus) of a com-
`plex number is
`lNl = Id + bil = [(a + bi)(a _ bi)]%
`= (a2 + bzfi
`
`(a+bi) + (c+dz')
`
`= (a+c)+(b+d)i
`(a+bi)—(c+di) = (a—c)+(b—d)i
`(a + bi)(c + dz)
`
`= (ac —~ bd) + (be + ad)i
`
`‘M
`
`Invensys Ex. 2019
`Micro Motion v. Invensys IPR 2014-00393, page 5
`
`

`

`magma”
`2-5i
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`(2.821?”
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`L.
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`.L.
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`_l
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`I
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`J
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`l
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`‘1 ‘ "’
`
`Figure 31
`
`
`
`.
`_
`'magmarv
`
`~-------------w
`
` Figure 132
`
`
`
`i.e., the modulus R and angle 9 are simply
`the polar coordinates of the point that rep-
`resents the number in the complex plane.
`Polar form is handy when complex num-
`bers have to be multiplied (or divided);
`you just multiply (divide) their magnitudes
`and add (subtract) their angles:
`
`(aeibxceid) : acei(b+d)
`
`Finally, to convert from polar to rectangu-
`lar form, just use Euler’s formula:
`
`“33'
`
`new 2 (1 cos b + to sin b
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`___________________________——————-———-—-——-—
`
`MATH REVIEW
`Appendix B
`
`1051
`
`You get them by writing out the number in
`the form a + bi, then taking either a or b.
`This may involve some multiplication or
`division, since the complex number may
`be a real mess.
`
`Jreal
`
`where R = (a2+b2)% and 9 = tan”1(b/a).
`This is usually written in a different way,
`., using the fact that
`a
`.
`.
`e = cos a: + 2 Sin 2;
`
`(You can easily derive the preceding result,
`known as Euler’s formula, by expanding
`the exponential in a Taylor series.) Thus
`we have the following equivalents:
`
`N : a + bi ___ Reie
`R = |N| = (NN*)% = (a2 + bzfi
`9 = tan—1(b/a)
`
`Complex numbers are sometimes repre-
`sented on the complex plane. It looks just
`If you have a complex number multiplying
`like an ordinary may graph, except that a
`a complex exponential, just do the neces-
`complex number is plotted by taking its
`sary multiplications. 1f
`real part as :1: and its imaginary part as
`.
`y;
`i.e., the axes represent REAL (11:) and
`N = a + 1"
`IMAGINARY (y), as shown in Figure Bl.
`Ne” = (a + bi)(cos z + isin 3:)
`In keeping with this analogy, you some-
`_
`_ b .
`)
`times see complex numbers written just
`— ((1 cos :1:
`smm
`like :5, y coordinates:
`+t(bcosa:+as1nm)
`a+bi+—>(a,b)
`
`
`Just as with ordinary any pairs, corn'plfif" DIFFERENTIATION (CALCULUS)
`
`
`numbers can be represented in polar coor-
`
`
`dmfteS; that’s 1.“)own as “magnitude, an- We start with the concept of a function
`
`
`g1e representation. For example-the num-
`f(ir),
`i.e., a formula that gives a value
`
`
`ber a + bi can also be written (Fig. B2)
`y = f (as) for each x. The function f(a:)
`
`a + bi" = (R, 9)
`should be singlevvalued i.e., it should give
`
`
`Note that "angle" above‘isthe "phase
`angle" or simply “phase."
`
`
`
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`Invensys EX. 2019
`.
`
`Micro Motion v. Invensys IPR 2014-00393, page 6
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